Another Question on Torsion Elements .... D&F Section 10.1, Exercise 8 ....

In summary, the exercise is asking for a rigorous solution to Exercise 8(c) of Section 10.1 in the book "Abstract Algebra" by Dummit and Foote. The solution involves considering a non-zero $R$-module and showing that it has non-zero torsion elements if certain conditions are met.
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I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ...

I am currently studying Chapter 10: Introduction to Module Theory ... ...

I need some help with an Exercise 8(c) of Section 10.1 ...

Exercise 8 of Section 10.1 reads as follows:https://www.physicsforums.com/attachments/8313Can someone please demonstrate a rigorous solution to 8(c) ...

Peter
 
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Hi Peter,

We know from the assumption that there are non-zero $a,b\in R$, such that $ab=0.$ Now let $M$ be a non-zero $R$-module. Note that we have

$M=\{m\in M: bm=0\}\cup \{m\in M: bm\neq 0\}.$

Since $M$ is a non-zero $R$-module, one of these two sets must contain a non-zero element. If the first contains said non-zero element, then $M$, by definition, has non-zero torsion elements since $b\neq 0.$ I will leave it to you to work on the case where $\{m\in M: bm=0\}=\{0\}.$
 
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FAQ: Another Question on Torsion Elements .... D&F Section 10.1, Exercise 8 ....

1. What are torsion elements in algebraic structures?

Torsion elements are elements in a group, ring, or module that have finite order, meaning that there exists some positive integer n such that multiplying the element by itself n times results in the identity element. In other words, the element "wraps around" itself in a cyclic manner.

2. How can torsion elements be identified in a given structure?

In a group, torsion elements can be identified by finding the smallest positive integer n such that the element raised to the n-th power is equal to the identity element. In a ring or module, torsion elements can be identified by finding the smallest positive integer n such that multiplying the element by itself n times yields the zero element.

3. Why are torsion elements important in algebraic structures?

Torsion elements play a significant role in understanding the structure of a group, ring, or module. They can provide information about the order and cyclic behavior of the elements in a structure, and can also help to identify substructures and factorization properties.

4. How are torsion elements related to torsion subgroups?

Torsion elements are the elements that make up a torsion subgroup, which is a subgroup consisting only of torsion elements. Torsion subgroups can provide insight into the structure of a larger group, and can also help to classify groups and determine their properties.

5. How can I use D&F Section 10.1 Exercise 8 to practice identifying torsion elements?

D&F Section 10.1 Exercise 8 provides a set of problems that require identifying torsion elements in various algebraic structures such as groups, rings, and modules. By working through these exercises, you can gain practice and improve your understanding of torsion elements and their properties in different contexts.

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